Let $X$ be any set. A $\textdef{permutation group}$ on $X$ is any subgroup of $\sym{X}$; that is, it is a subset of $\sym{X}$ that contains the identity and is closed under composition and taking inverses.

<wrap help> This definition requires that we first define the symmetric group, and that we have defined subgroup, and that we have developed some intuition about how to something is a subgroup. I would prefer that we change this. </wrap>

Let $X$ be any set. A permutation on $X$ is a bijection on $X$. A $\textdef{permutation group}$ on $X$ is a set of bijections on $X$ that ….

• $\sym{X}$ denotes the symmetric group on $X$.

definition.permutation_group.tex

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% DEPENDENCIES 
% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}}  \DeclareMathOperator{\sym}{Sym} 
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\begin{definition}
Let $X$ be any set. A \textdef{permutation group} on $X$ is any subgroup of $\sym{X}$; that is, it is a subset of $\sym{X}$ that contains the identity and is closed under composition and taking inverses.
\end{definition}

• Permutation group

definition needsreview rjosh